Residuals for Fitted Point Process Model
Given a point process model fitted to a point pattern, compute residuals.
## S3 method for class 'ppm'
residuals(object, type="raw", ...,
check=TRUE, drop=FALSE,
fittedvalues=NULL,
new.coef=NULL, dropcoef=FALSE,
quad=NULL)object |
The fitted point process model (an object of class |
type |
String indicating the type of residuals to be calculated.
Current options are
|
... |
Ignored. |
check |
Logical value indicating whether to check the internal format
of |
drop |
Logical value determining whether to delete quadrature points
that were not used to fit the model. See |
fittedvalues |
Vector of fitted values for the conditional intensity at the quadrature points, from which the residuals will be computed. For expert use only. |
new.coef |
Optional. Numeric vector of coefficients for the model,
replacing |
dropcoef |
Internal use only. |
quad |
Optional. Data specifying how to re-fit the model.
A list of arguments passed to |
This function computes several kinds of residuals for the fit of
a point process model to a spatial point pattern dataset
(Baddeley et al, 2005).
Use plot.msr to plot the residuals directly,
or diagnose.ppm
to produce diagnostic plots based on these residuals.
The argument object must be a fitted point process model
(object of class "ppm"). Such objects are produced by the maximum
pseudolikelihood fitting algorithm ppm.
This fitted model object contains complete
information about the original data pattern.
Residuals are attached both to the data points and to some other points in the window of observation (namely, to the dummy points of the quadrature scheme used to fit the model). If the fitted model is correct, then the sum of the residuals over all (data and dummy) points in a spatial region B has mean zero. For further explanation, see Baddeley et al (2005).
The type of residual
is chosen by the argument type. Current options are
"raw":the raw residuals
r[j] = z[j] - w[j] lambda[j]
at the quadrature points u[j], where z[j] is the indicator equal to 1 if u[j] is a data point and 0 if u[j] is a dummy point; w[j] is the quadrature weight attached to u[j]; and
lambda[j] = lambda(u[j],x)
is the conditional intensity of the fitted model at u[j]. These are the spatial analogue of the martingale residuals of a one-dimensional counting process.
"inverse":the ‘inverse-lambda’ residuals (Baddeley et al, 2005)
rI[j] = r[j]/lambda[j] = z[j]/lambda[j] - w[j]
obtained by dividing the raw residuals by
the fitted conditional intensity. These are
a counterpart of the exponential energy marks (see eem).
"pearson":the Pearson residuals (Baddeley et al, 2005)
rP[j] = r[j]/sqrt(lambda[j]) = z[j]/sqrt(lambda[j]) - w[j] sqrt(lambda[j])
obtained by dividing the raw residuals by the square root of the fitted conditional intensity. The Pearson residuals are standardised, in the sense that if the model (true and fitted) is Poisson, then the sum of the Pearson residuals in a spatial region B has variance equal to the area of B.
"score":the score residuals (Baddeley et al, 2005)
r[j] = (z[j] - w[j] lambda[j]) * x[j,]
obtained by multiplying the raw residuals r[j] by the covariates x[j,] for quadrature point j. The score residuals always sum to zero.
The result of residuals.ppm is a measure
(object of class "msr").
Use plot.msr to plot the residuals directly,
or diagnose.ppm to produce diagnostic plots
based on these residuals.
Use integral.msr to compute the total residual.
By default,
the window of the measure is the same as the original window
of the data. If drop=TRUE then the window is the
domain of integration of the pseudolikelihood or composite likelihood.
This only matters when the model object was fitted using
the border correction: in that case, if drop=TRUE the
window of the residuals is the erosion of the original data window
by the border correction distance rbord.
An object of class "msr"
representing a signed measure or vector-valued measure
(see msr). This object can be plotted.
Sometimes we want to modify the calculation of residuals by using
different values for the model parameters. This capability is
provided by the arguments new.coef and quad.
If new.coef is given, then the residuals will be computed
by taking the model parameters to be new.coef.
This should be a numeric vector
of the same length as the vector of fitted model parameters
coef(object).
If new.coef is missing and quad is given,
then the model parameters will
be determined by re-fitting the model using a new
quadrature scheme specified by quad.
Residuals will be computed for the
original model object using these new parameter values.
The argument quad should normally be
a list of arguments in name=value format that will be
passed to quadscheme (together with
the original data points) to determine the new quadrature scheme.
It may also be a quadrature scheme (object of class
"quad") to which the model should be fitted, or a
point pattern (object of class "ppp") specifying the
dummy points in a new quadrature scheme.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner r.turner@auckland.ac.nz
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
fit <- ppm(cells, ~x, Strauss(r=0.15)) # Pearson residuals rp <- residuals(fit, type="pe") rp # simulated data X <- rStrauss(100,0.7,0.05) # fit Strauss model fit <- ppm(X, ~1, Strauss(0.05)) res.fit <- residuals(fit) # check that total residual is 0 integral.msr(residuals(fit, drop=TRUE)) # true model parameters truecoef <- c(log(100), log(0.7)) res.true <- residuals(fit, new.coef=truecoef)
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